Space of modular forms of level 304, weight 4, and trivial character

• Label: 304.4.a
• Level: $304$
• Weight: $4$
• Character orbit: 304.a
• Rep. character: $\chi_{304}(1,\cdot)$
• Character field: $\Q$
• Dimension: $27$
• Newform subspaces: $11$
• Sturm bound: $160$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 304 = 2^{4} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	304.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 11 \)
Sturm bound:			\(160\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(304))\).

	Total	New	Old
Modular forms	126	27	99
Cusp forms	114	27	87
Eisenstein series	12	0	12

The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators and the Fricke involution.

\(2\)	\(19\)	Fricke		Total				Cusp				Eisenstein
				All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)		\(33\)	\(8\)	\(25\)		\(30\)	\(8\)	\(22\)		\(3\)	\(0\)	\(3\)
\(+\)	\(-\)	\(-\)		\(30\)	\(5\)	\(25\)		\(27\)	\(5\)	\(22\)		\(3\)	\(0\)	\(3\)
\(-\)	\(+\)	\(-\)		\(30\)	\(7\)	\(23\)		\(27\)	\(7\)	\(20\)		\(3\)	\(0\)	\(3\)
\(-\)	\(-\)	\(+\)		\(33\)	\(7\)	\(26\)		\(30\)	\(7\)	\(23\)		\(3\)	\(0\)	\(3\)
Plus space		\(+\)		\(66\)	\(15\)	\(51\)		\(60\)	\(15\)	\(45\)		\(6\)	\(0\)	\(6\)
Minus space		\(-\)		\(60\)	\(12\)	\(48\)		\(54\)	\(12\)	\(42\)		\(6\)	\(0\)	\(6\)

Trace form

27 * q + 2 * q^5 + 14 * q^7 + 243 * q^9 - 86 * q^11 - 46 * q^13 + 12 * q^15 - 26 * q^17 - 57 * q^19 + 136 * q^21 - 68 * q^23 + 865 * q^25 + 576 * q^27 - 398 * q^29 - 24 * q^33 - 246 * q^35 + 322 * q^37 - 540 * q^39 + 118 * q^41 - 230 * q^43 - 350 * q^45 + 1262 * q^47 + 915 * q^49 + 1012 * q^51 + 90 * q^53 + 830 * q^55 - 272 * q^59 - 390 * q^61 + 610 * q^63 - 348 * q^65 + 1020 * q^67 + 24 * q^69 + 1124 * q^71 - 1354 * q^73 + 1068 * q^75 + 1608 * q^77 + 1080 * q^79 + 1571 * q^81 - 1676 * q^83 - 1588 * q^85 - 3540 * q^87 - 314 * q^89 + 952 * q^91 - 2248 * q^93 + 760 * q^95 - 290 * q^97 + 1154 * q^99

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(304))\) into newform subspaces

Label	Level	Weight	Char	Prim	Char order	Dim	Rel. Dim	$A$	Field	CM	Self-dual	Twist minimal	Largest	Maximal	Minimal twist	Inner twists	Rank*	Traces					A-L signs		Fricke sign	Coefficient ring index	Sato-Tate	$q$-expansion
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		2	19
304.4.a.a	$304$	$4$	304.a	1.a	$1$	$1$	$1$	$17.937$	\(\Q\)	None	✓				38.4.a.a	$1$	$1$	\(0\)	\(2\)	\(-9\)	\(31\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{3}-9q^{5}+31q^{7}-23q^{9}-57q^{11}+\cdots\)
304.4.a.b	$304$	$4$	304.a	1.a	$1$	$1$	$1$	$17.937$	\(\Q\)	None	✓				19.4.a.a	$1$	$1$	\(0\)	\(5\)	\(-12\)	\(-11\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+5q^{3}-12q^{5}-11q^{7}-2q^{9}+54q^{11}+\cdots\)
304.4.a.c	$304$	$4$	304.a	1.a	$1$	$2$	$2$	$17.937$	\(\Q(\sqrt{73}) \)	None	✓				38.4.a.c	$1$	$1$	\(0\)	\(-9\)	\(-9\)	\(18\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-4-\beta )q^{3}+(-3-3\beta )q^{5}+(7+4\beta )q^{7}+\cdots\)
304.4.a.d	$304$	$4$	304.a	1.a	$1$	$2$	$2$	$17.937$	\(\Q(\sqrt{177}) \)	None	✓				38.4.a.b	$1$	$0$	\(0\)	\(-1\)	\(10\)	\(-57\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta q^{3}+(6-2\beta )q^{5}+(-28-\beta )q^{7}+\cdots\)
304.4.a.e	$304$	$4$	304.a	1.a	$1$	$2$	$2$	$17.937$	\(\Q(\sqrt{57}) \)	None	✓				152.4.a.a	$1$	$1$	\(0\)	\(1\)	\(-5\)	\(6\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{3}+(-3+\beta )q^{5}+(5-4\beta )q^{7}+\cdots\)
304.4.a.f	$304$	$4$	304.a	1.a	$1$	$2$	$2$	$17.937$	\(\Q(\sqrt{33}) \)	None	✓				76.4.a.a	$1$	$0$	\(0\)	\(5\)	\(-5\)	\(30\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(3-\beta )q^{3}-5\beta q^{5}+(17-4\beta )q^{7}+\cdots\)
304.4.a.g	$304$	$4$	304.a	1.a	$1$	$3$	$3$	$17.937$	3.3.7057.1	None	✓		✓		152.4.a.c	$1$	$1$	\(0\)	\(-4\)	\(7\)	\(-7\)		$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(-1-\beta _{2})q^{3}+(2+2\beta _{1}+\beta _{2})q^{5}+\cdots\)
304.4.a.h	$304$	$4$	304.a	1.a	$1$	$3$	$3$	$17.937$	3.3.35529.1	None	✓		✓		76.4.a.b	$1$	$1$	\(0\)	\(-1\)	\(9\)	\(-44\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-\beta _{1}-\beta _{2})q^{3}+(3+\beta _{2})q^{5}+(-15+\cdots)q^{7}+\cdots\)
304.4.a.i	$304$	$4$	304.a	1.a	$1$	$3$	$3$	$17.937$	3.3.3144.1	None	✓		✓		19.4.a.b	$1$	$0$	\(0\)	\(-1\)	\(14\)	\(35\)		$-$	$-$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q-\beta _{2}q^{3}+(4-\beta _{1}+\beta _{2})q^{5}+(11-2\beta _{1}+\cdots)q^{7}+\cdots\)
304.4.a.j	$304$	$4$	304.a	1.a	$1$	$3$	$3$	$17.937$	3.3.3221.1	None	✓				152.4.a.b	$1$	$0$	\(0\)	\(5\)	\(2\)	\(35\)		$+$	$+$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q+(2+\beta _{2})q^{3}+(1-\beta _{1}+2\beta _{2})q^{5}+(11+\cdots)q^{7}+\cdots\)
304.4.a.k	$304$	$4$	304.a	1.a	$1$	$5$	$5$	$17.937$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓		✓		152.4.a.d	$1$	$0$	\(0\)	\(-2\)	\(0\)	\(-22\)		$+$	$+$	$+$	$2^{3}$	$\mathrm{SU}(2)$	\(q-\beta _{2}q^{3}+(\beta _{1}+\beta _{2})q^{5}+(-4-\beta _{2}+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(304))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(304)) \simeq \) \(S_{4}^{\mathrm{new}}(\Gamma_0(8))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(16))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(19))\)\(^{\oplus 5}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(38))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(76))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(152))\)\(^{\oplus 2}\)